SUNY Geneseo Department of Mathematics
Friday, September 11
Math 221 02
Fall 2020
Prof. Doug Baldwin
Due dates for problem sets. Ideally, there are general due dates for the problem sets, reflected in the dates I put at the top of each. In particular, the dates indicate when I plan for you to be doing the problem set, in the broad sense of “doing” — i.e., reading over it, thinking about the problems, asking questions about them, actually writing out answers, and grading. So it’s nominally due towards the end of that range of dates (if you have a weekly meeting at that time), or shortly after (if your weekly meeting is nearer the beginning of the dates on the problem set). But in the spirit of mastery grading, there’s no grade or other penalty for not meeting these deadlines, i.e., they won’t be enforced — except that maybe I’ll start nagging you if you fall very much behind them.
Videos of the SI sessions should be coming soon.
The next session is Sunday 6:00 - 7:30. See Emiliana’s announcement for the Zoom link.
Based on “Additional Limit Evaluation Techniques” in section 2.3 of the textbook, and the algebraic methods discussion:
Consider the limit
If you try to evaluate this by just plugging x = 1 into the expression (i.e., using limit laws), you find that the numerator and denominator are both zero, and you can’t divide 0 by 0. This sort of problem happens a lot when finding limits.
One common way of dealing with the problem is to factor either the numerator or the denominator (or both) in order to remove the division by 0 by canceling out part or all of the denominator. So the general procedure looks like...
In the case of this example, this procedure looks like this:
Here’s a limit that doesn’t obviously lend itself to factoring:
Another trick that is often useful for simplifying fractions involving square roots is “multiplying by the conjugate.” The “conjugate” of an expression a + b is a - b, and vice versa. The reason multiplying a sum or difference involving a square root by its conjugate can help is related to the factoring rule for a difference of squares:
meaning that multiplying a sum or difference by its conjugate will leave you with a difference of squares; if one of the original terms involved a square root, squaring it will eliminate the root.
Of course you can’t just multiply a sum or difference by its conjugate without changing its value. To avoid this, multiply by a fraction consisting of the conjugate over itself — such a fraction is equal to one, so multiplying by it doesn’t change values. As with factoring, the goal in this multiplication is to get a situation where something cancels out between the numerator and denominator, once again eliminating a divide-by-0 problem.
The overall strategy for dealing with a fraction whose numerator or denominator involves a sum or difference in which one term is a square root looks like this:
We used this strategy as follows to solve the above limit:
Try finding this limit:
Everyone in class used multiplying by the conjugate, which is a good way to solve this:
But you can also do this by factoring, if you think of the denominator as a difference of squares:
This approach ends up in the same place the first one did, which is a good thing (math should come out the same even when you do it different ways). This example is a nice illustration of how you can sometimes solve problems in either of several different ways.
Another sort of limit that comes up from time to time involves a built-up fraction (i.e., a fraction composed of other fractions). For example
The general strategy in such cases involves trying to simplify the fraction to the point where you can use limit laws to evaluate the limit. Two useful rules for simplifying fractions involve putting sums or differences of fractions over common denominators, and combining denominators and/or numerators via the rules
The goal in using these rules is to simplify the built-up fraction down a single fraction, i.e., one with no other fractions in its numerator or denominator, then cancel like terms in the numerator or denominator to eliminate division by 0. The strategy for doing so isn’t as clear-cut as the ones for factoring or multiplying by a conjugate, but it looks roughly like this. You might find, though, that not all steps apply to all fractions, or that you might need to repeat some steps several times.
Here’s how we used this strategy on the example:
As a final example, consider
As with the other examples we’ve done, we need to simplify this limit to the point where applying limit laws won’t entail division by 0. Because this example involves trig functions, a good way to do that is to use trig identities to rewrite the numerator or denominator and try to get the problematic division by 0 to cancel away. The general strategy looks like:
The last step, “apply limit laws,” is a little bit beyond where we are with limits in this course, since we don’t officially know that there are limit laws for trigonometric (or other) functions. But we’ll soon see how you can extend the limit laws for arithmetic to at least some functions, including the one in this example.
In the example, the process looks like this:
There is a new problem set dealing with limit laws and algebra. Work on it next week. Grade it late next week if your appointment comes at a suitable time, or early the week after.
See the handout for more details.
Evaluating limits in Mathematica.
One-sided limits, i.e., limits that hold only as you approach from one side.
Read “One-Sided Limits” in section 2.2 of the textbook.
Watch for a discussion of them to be announced soon.
Next class (Monday) is a cohort C class.